How to Calculate the Tangential Component of the Coriolis Force on a Carousel?

[sandy.bridge]

Homework Statement

Hello all,
I was hoping to get some help on this difficult question I encountered. Thanks in advance.
Cheers.

"Imagine you are taking a ride on a horizontally spinning carousel. Suppose that at a certain moment, you are moving with linear speed $\vec{v}$ and

$\frac{d}{dt}\vec{v}=\vec{b}+.5\vec{v}$ where $\vec{b}⋅\vec{v}=0$.

Let $\vec{C_t}(t)$ denote the tangential component of the Coriolis force exerted on your body at time t and let its magnitude equate 1. Calculate

$\frac{d}{dt}|\vec{C_t}|^2(t)$
"

The normal component of angular velocity is constant.

The Attempt at a Solution

$\vec{C_t}(t)=-2\vec{\Omega}\times\vec{v}$

I really don't know where to even begin. I can't seem to get any questions regarding the Coriolis effect.

 

[zhermes]

What are the relevant equations for the coriolis force?

 

[sandy.bridge]

$\vec{C}=-2\vec{\Omega}_T\times\vec{v}-2\vec{\Omega}_N\times\vec{v}$, however, we are only interested in the tangential component, no?

We know $|-2\vec{\Omega}_T\times\vec{v}|=1$ and $\vec{\Omega}_T⋅\vec{\Omega_N}=0$

 

[zhermes]

Sorry---I didn't see the equation you had under 'attempt at a solution' previously.
Why do you have two angular velocities?

If you're initially with a linear velocity $\vec{v}$, what direction are $\vec{v}$ and $\vec{b}$ in?

If you find the total coriolis force, how can you then reduce it to only the tangential component?

 

[sandy.bridge]

There are two angular velocities, one tangential, denoted T, and one normal, denoted N. This is the equation that my textbook provides. The velocity vector is tangent to the circular motion of the carousel, and the b vector either is directed to the center of the carousel or away from the center.

I'm really not getting much further with this.

 

[zhermes]

There are two angular velocities, one tangential, denoted T, and one normal, denoted N. This is the equation that my textbook provides. The velocity vector is tangent to the circular motion of the carousel, and the b vector either is directed to the center of the carousel or away from the center.

I'm really not getting much further with this.

There's only one component to the carousel's angular velocity. You're right about the direction of v and b. So if you have the total coriolis force, what vector operations can you use (along with the vectors v and b) to find only the tangential component. it might help to draw a diagram.

 

[sandy.bridge]

Are we supposed to assume that vector b is the position vector?

$|-2\vec{\Omega}\times\vec{v}|=1$

 

[zhermes]

No... What does the cross product do? What about the dot product?

 

[sandy.bridge]

Cross product produces a vector perpendicular to the two vectors. In my textbook that is how the write the formula for the Coriolis force.

 

[zhermes]

Let me make this more clear: you can use the dot-product, and or cross-product to find the component of the coriolis force that you're looking for.

 

[sandy.bridge]

I'm not getting this question at all.. Which direction is the force vector going to be pointing? Will it be tangent to the carousel, parallel with the velocity vector?

 

[zhermes]

$$C \propto \Omega \times v$$
$\Omega$ points perpendicularly upwards from the carousel.
So what direction is C in, relative to $\Omega$ and v?

 

[sandy.bridge]

C will be directed radially inward? If that's the case, then the dot product of angular velocity and the C force will equal zero?

 

[sandy.bridge]

Scratch that, the force will be directed out